Many manufacturing operations involve the manufacture of discrete parts. Semiconductor fabrication includes the fabrication of discrete parts, such as runs or batches of wafers, each of which includes one or more wafers fabricated using similar processing. One challenge in semiconductor fabrication is controlling equipment inputs from run to run. The challenge in run-to-run control stems from a lack of real-time information about process states and output states while processing a given wafer or group of wafers and variable input states of incoming wafers.
Chemical-mechanical polishing (CMP) is a common and rapidly growing process used in the fabrication of semiconductor wafers for planarizing silicon dioxide as well as other types of layers on semiconductor wafers. Chemical mechanical polishing typically utilizes an abrasive slurry disbursed in an alkaline or acidic solution to planarize the surface of the wafer through a combination of mechanical and chemical action. A typical chemical mechanical polishing tool includes a rotatable circular platen or table on which a polishing pad is mounted and a polishing device is positioned above the pad. The polishing device includes one or more rotating carrier heads to which wafers can be secured typically through the use of vacuum pressure. In use, the platen is rotated and an abrasive slurry is disbursed onto the polishing pad. Once the slurry has been applied to the polishing pad, a downforce is applied to each rotating carrier head to press its wafer against the polishing pad. As the wafer is pressed against the polishing pad, the surface of the wafer is mechanically and chemically polished. In a manufacturing operation, the usual adjustment parameter or tool input is polishing time (though in some operations, carrier downforce is input as well). Other parameters, such as table speed and carrier downforces are fixed for the process.
As semiconductor devices are scaled down, the importance of run-to-run control increases. In particular, it becomes increasingly important to control tool outputs as variations in tool outputs deleteriously impact subsequent fabrication steps and degrade device performance. In CMP, for example, tool outputs, such as post-polish thicknesses of polished layers, must be accurately controlled as variations in post-polish thicknesses can, for example, significantly degrade subsequent processing steps, such as lithography.
Conventional run-to-run control of manufacturing tools, such as CMP tools, typically involves the use of Exponentially Weighted Moving Average (EWMA) controllers. EWMA controllers typically use a linear regression process model such as: EQU y.sub.k =B*u.sub.k.vertline.k-1 +c.sub.k.vertline.k-1 +e.sub.k [1]
Where y.sub.k is the output at batch k, B is the process gain, u.sub.k.vertline.k-1 is the input at batch k calculated from information up through batch k-1, c.sub.k.vertline.k-1 is the estimate for the intercept, and e.sub.k is unknown process noise entering the system. Typically, the system gain and the initial value of the intercept is modeled a priori from designed experiments.
The intercept is typically updated recursively by an observer of the form: EQU c.sub.k.vertline.k-1 =.lambda.*(y.sub.k-1 -B*u.sub.k-1.vertline.k-2)+(1-.lambda.)*c.sub.k-1.vertline.k-2 [2]
where .lambda. is the exponential weighting factor, or tuning parameter, of the observer. The weighting factor .lambda. takes a value between 0 and 1 and is chosen based on the desired properties of the observer. Using the updated intercept, the input for batch k (u.sub.k.vertline.k-1) is determined from the following relationship: EQU u.sub.k.vertline.k-1 =(T-c.sub.k.vertline.k-1)/B [3]
where T is a target output thickness.
Other controllers, such as predictor-corrector controllers (PCC), have been suggested for run-to-run control. PCC uses a second exponential filter to forecast the trend in the estimated intercept in an attempt to predict how the intercept will change in the future, while the standard EWMA controller simply assumes the intercept will remain constant. The additional observer equation of the PCC is of the form: EQU .delta.c.sub.k.vertline.k-1 =.gamma.*(y.sub.k -B*u.sub.k.vertline.k-1 -c.sub.k.vertline.k-1)+(1-.gamma.)*.delta.c.sub.k-1.vertline.k-2 [4]
where .delta.c.sub.k.vertline.k-1 is the smoothed trend of the intercept. After this modification, the model prediction equation then becomes: EQU .gamma..sub.k =B*u.sub.k.vertline.k-1 +c.sub.k.vertline.k-1 [5]
with the new prediction for the intercept given by: EQU c.sub.k.vertline.k-1 =c.sub.k.vertline.k-1 +.delta.c.sub.k.vertline.k-1 [6]
Despite the use of EWMA and PCC controllers, variations in tool output still significantly impact semiconductor fabrication and device performance. The significance of the impact is also growing as a result of the ever decreasing size of semiconductor devices. Consequently, semiconductor manufacturers seek systems and methods for more accurately controlling the manufacture of semiconductor wafers.